An ordered Sylow tower of complexion $\mathrm{\gamma }$ for a finite group $G$ is a normal series

such that, for each $i$, the quotient group $\frac{G_i}{G_{i+1}}$ is isomorphic to a Sylow $p_i$-subgroup of $G$, for some prime $p_i$, and such that $p_1$, $p_2$, ..., $p_r$  are all the distinct prime divisors of the order of $G$, and occur in the same order as they do in the list gamma.

A finite group may, or may not, have an ordered Sylow tower of a specified complexion. If it has an ordered Sylow tower of one complexion, it may not have an ordered Sylow tower for a different complexion.

Every finite nilpotent group has a Sylow tower (of every possible complexion), and a finite group with a Sylow tower (of any complexion) is necessarily soluble.

An ordered Sylow tower group is, of course, a Sylow tower group. (See SylowTower.)

The OrderedSylowTower( G, 'complexion' = gamma ) command computes an ordered Sylow tower of compexion gamma for the group G if one exists. The returned Sylow tower is an object of type NormalSeries.

In addition to the methods available for any Series object, a Sylow tower T also supports the Complexion( T ) method, which returns the complexion of the computed tower, as a list of primes.

The IsOrderedSylowTowerGroup( G, 'complexion' = gamma ) command returns true if G has a Sylow tower of complexion gamma, and returns false if not.

Both OrderedSylowTower and IsOrderedSylowTowerGroup can be called without the complex = gamma option, in which case the default complexion used is the list of all the prime divisors of the order of the group G in descending order. (An ordered Sylow tower of this complexion is sometimes called an ordered Sylow tower of supersoluble type.)

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$\mathrm{seq}\left(H\,H=\mathrm{OrderedSylowTower}\left(G\right)\right)$

Error, group GroupTheory:-AlternatingGroup(4) has no ordered Sylow tower of complexion [3, 2]

$T≔\mathrm{OrderedSylowTower}\left(G\,'\mathrm{complexion}'=\left[2\,3\right]\right)$

$T≔⟨⟩◃⟨\left(1\,3\right)\left(2\,4\right)\,\left(1\,2\right)\left(3\,4\right)⟩◃⟨\left(1\,3\right)\left(2\,4\right)\,\left(1\,2\right)\left(3\,4\right)\,\left(2\,3\,4\right)⟩$

$\mathrm{type}\left(T\,'\mathrm{NormalSeries}'\right)$

$\mathrm{true}$

$\mathrm{seq}\left(\mathrm{GroupOrder}\left(H\right)\,H=T\right)$

$1,4,12$

$\mathrm{seq}\left(\mathrm{Index}\left(T\left[i-1\right]\,T\left[i\right]\right)\,i=2..:-\mathrm{numelems}\left(T\right)\right)$

$4,3$

$\mathrm{Complexion}\left(T\right)$

$\left[2\,3\right]$

$\mathrm{IsOrderedSylowTowerGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$

$\mathrm{true}$

$\mathrm{IsOrderedSylowTowerGroup}\left(\mathrm{DihedralGroup}\left(5\right)\,'\mathrm{complexion}'=\left[5\,2\right]\right)$

$\mathrm{true}$

$\mathrm{IsOrderedSylowTowerGroup}\left(\mathrm{DihedralGroup}\left(5\right)\,'\mathrm{complexion}'=\left[2\,5\right]\right)$

$\mathrm{false}$

The GroupTheory[OrderedSylowTower] and GroupTheory[IsOrderedSylowTowerGroup] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.